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The Role of Sine and Cosine Components in Love Waves and Rayleigh Waves for Energy Hauling During Earthquake

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The Role of Sine and Cosine Components in Love Waves and Rayleigh Waves for Energy Hauling During Earthquake American Journal of Applied Sciences 7 (12): 1550-1557, 2010 ISSN 1546-9239 © 2010 Science Publications The Role of Sine and Cosine Components in Love Waves and Rayleigh Waves for Energy Hauling during Earthquake Zainal Abdul Aziz and Dennis Ling Chuan Ching Department of Mathematics, Faculty of Science, University Technology Malaysia, 81310 Johor, Malaysia Abstract: Problem statement: Love waves or shear waves are referred to dispersive SH waves. The diffusive parameters are found to produce complex amplitude and diffusive waves for energy hauling. Approach: Analytical solutions are found for analysis and discussions. Results: When the standing waves are compared with the Love waves, similar cosine component is found in both waves. Energy hauling between waves is impossible for Love waves if the cosine component is independent. In the view of 3D particle motion, the sine component of the Rayleigh waves will interfere with shear waves and thus the energy hauling capability of shear waves are formed for generating directed dislocations. Nevertheless, the cnoidal waves with low elliptic parameter are shown to act analogous to Rayleigh waves. Conclusion: Love waves or Rayleigh waves are unable to propagate energy once the sine component does not exist. Key words: Energy hauling, sine and cosine, amplitude, love waves, Rayleigh waves, Cnoidal waves, earthquake, phase velocity, periodic, isotropic medium, illustrated, anisotropic INTRODUCTION seismology, Love waves are referred to shear waves and SH waves in layered medium are similar to Love waves. For this reason, the similarity between standing Seismic body waves are classified to P, SH and SV waves. Similar waves have being derived in vector waves and Love waves must be exploited. Complexity form (Ari-Menahem and Singh, 1981). These waves of fractures and Love waves must be explained to will give the surface Rayleigh waves and Loves waves resolve the mechanism of vein structures formation that decay exponentially with depth. Rayleigh waves especially the direction of dislocation. and Love waves are recognized for near surface Standing waves and progressive waves are displacement and there are categorized under dispersive different in general. Standing waves do not involve in waves since the phase and group velocities are different the form of energy and momentum transfer between (Bhatnagar, 1979). Tsunami waves or long waves are waves separated by nodes while progressive waves are also the surface waves. However, these waves are propagating the energy. Since Rayleigh waves are near dissimilar with Rayleigh waves or Love waves (Ribal, surface waves, multi layer mediums are not suitable for 2007; 2008). showing near surface displacement. However, the Standing waves are known as stationary waves that splitting shear waves or SV waves from Rayleigh do not propagate. There arise in a motionless medium waves have affected the Rayleigh wave’s dispersive as a result of interference involving two waves traveling behavior for layered anisotropic medium (Zhang et al ., in opposite directions. Similar incidence can be 2009). The statement for splitting SV waves is explained through the movement along the tectonic progressive in layered medium must be explained. plate where the interference of waves is invincible. The Energy hauling capability should be considered as well. capability of standing waves to propagate forward has Sine and cosine components in progressive waves attracted researchers’ interest. In experimental study, and standing waves explain the different between (Brothers et al ., 1996) attributed the standing waves Rayleigh and Love waves. Seismic waves are periodic from systematic fractures such as vein structures during waves and thus the displacement has a periodic seismic wave shaking. Later, Tsuneo and Yujiro variation with time and distance. Despite being (Ohsumi and Ogawa, 2008) found vein structures that periodic, conidial waves have sharper crest and flatter are formed by shear waves during earthquake. In troughs than sinusoidal waves. Usually, these waves are Corresponding Author: Zainal Abdul Aziz, Department of Mathematics, Faculty of Science, University Technology Malaysia, 81310 Johor, Malaysia 1550 Am. J. Applied Sci., 7 (12): 1550-1557, 2010 referred to ocean waves which are distinguished by the Where: elliptic parameter. When the elliptic parameter reduces C = The velocity of the phase to zero, smooth repetitive oscillations are formed and δ = α or β = The velocity of the P or S waves these waves are similar to seismic waves. Both waves respectively should look alike at medium surface if the antinodes of K = Refers to the wavenumbers the waves are ignored. Eventually, it must be explained η η η the Rayleigh wave has a possibility to transform into δ = α or β = Known as diffusion parameter Cnoidal waves once the elliptic parameter is increasing to produce nonlinear seismic waves that can propagate Inserting the relations (4) and (5) into equations energy. Apparently, nonlinear waves are known for self (1), (2) and (3), we will get the dispersive form of reinforcing and capable of propagating vast energy in seismic waves as: the medium (Dauxious and Peyrard, 2006; Drazin and Johnson, 1989). U=± Aa( na) expikct[ ( −η x∓ z )] (6) P xα z α MATERIALS AND METHODS U= Ba( ∓ na) expikct[ ( − x ∓ η z )] (7) Problem statement: Seismic waves are divided into SV zβ x β two groups P and S in according to the propagation direction. The S waves again are categorized into two U= Ca exp[ ik( ct − x∓ η β z )] (8) sub groups SV and SH waves. The explanations for SH y seismic waves in energy hauling and 3D particle motion are more remarkable if these waves are featured in sine The combination of Eq. 6 and 7 will form the and cosine components. Rayleigh waves while Eq. 8 alone is known as Love The combination of the Loves waves and Rayleigh waves (Ben-Menahem and Singh, 1981). These dispersive waves will give the 3D particle motion as shown in waves will decay exponentially with finite depth. Fig. 1. P, SV and SH waves introduced by Ben- For the case of δ>c, we get the complex ηδ. Menahem and Singh (1981) read: Inserting the complex ηδ to either Eq. 6-8, the selected lx+ nz Eq. 8 reads: U= A() la + na exp iw t − (1) P x z α U= Cexp( i ηβ z) exp[] ik() ct − x (9) lx+ nz SH U=−+ B() na la exp iw t − (2) SV x z β After linear polarization with respect to y-direction. C lx+ nz Is the amplitude while exp(i ηβz) is referring to the U= Ca expiw t − (3) SH y β diffusive term. Apparently, Eq. 9 is non diffusive since the term exp(i ηβz) is in complex (Bhatnagar, 1979). Where: l and n = The vector components Refer to the unit vector ax, a y and a z = α and β = The velocity for P and S waves A, B and C = Refer to the amplitude of the waves with finite depth z and w = The frequency of the waves For elliptic particle motion, similar frequency with respect to the wave is expected. It is appropriate to reduce (1), (2) and (3) through the relations that read: c2 δ nδ = − 1, c = (4) δ2 l c2 x± − 1 z 2 lx+ nz x + nzl δ = = (5) δ δ c l Fig. 1: The seismic wave’s framework 1551 Am. J. Applied Sci., 7 (12): 1550-1557, 2010 components introduced by Ben-Menahem and Singh (1981), the stress read: τ=µu =µ[ Cexp( −η i kz) + Cexpi( η kz )] 32 2,311β 2 β (11) exp[] ik() ct− x τ=µu =µ expikct[ ( − x )] (12) 32 2,3 1 where, u 1, u2, u 3 are in x, y, z directions. Equation 11 shows the stress within the layered medium while the Fig. 2: The amplitude and diffusion behavior of the S free surface stress is depicted through Eq. 12 when z = 0. waves Despite the absent of complex amplitude and diffusive term, the periodic sinusoidal waves are still exist. Love waves are usually explained in a layer over a half space medium. There is significant different between free surface stress and τ stress. Stress is also 32 recognized as diffusion energy. Free surface has less energy at the free surface whereas τ32 stress produces radical stress in close proximity to the surface featuring complex or non amplitude as illustrated in Fig. 4a and 4b respectively. Equation 10 can be written in cosine form with amplitude C1 = C 2 that reads: Fig. 3: The interference of S waves U= Ca cos( ηβ kh) exp[] ik() ct − x (13) SH y From Eq. 6-8, ηα and ηβ give the dispersive waves with amplitude A, B and C. The explanation can go into Equation 13 shares the similarity with the standing diffusive characteristic of the waves and also the waves (Bhatnagar, 1979). complex amplitudes. According to (Bhatnagar, 1979), the standing Complex amplitude is usually referred to phasor or waves do not include in the form of energy and amplitude of the waves. Figure 2 illustrates the momentum transfer between waves separated by nodes. propagation behavior of the SV waves featuring This explains Love waves do not forward energy. Since complex amplitude. It is possible for SV waves become Love waves only contribute to the cosine component, SH waves after polarization. Conforming to the rotating this kind of waves are considered as non moving waves behavior, S wave is plotted for the dispersion and which vibrate at the origin without participating in diffusion behaviors as illustrated in Fig. 3. S waves energy transformation. Conforming to the definition of have dispersed into many and diffused with depth z. standing waves, the capability of Love waves in energy These S waves are not featuring energy hauling behavior. hauling is non-existence.
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